Independence, Domination, Irredundance, and Forbidden Pairs

For different properties \(\mathcal{P}\) of a connected graph \(G\), we characterize the connected graphs \(F\) (resp. the pairs \((X,Y)\) of connected graphs) such that \(G\) has Property \(\mathcal{P}\) if \(G\) does not admit \(F\) (resp. neither \(X\) nor \(Y\)) as an induced subgraph.

We consider here the lower independence, domination, and irredundance parameters, which are related by the well-known inequalities \(ir \leq \gamma \leq i \leq \alpha \leq \Gamma \leq IR\), and the properties \(\mathcal{P}\) correspond to the equality between some
of these parameters.